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''of a topological space $X$, base of a topology, basis of a topology, open base''

A family $\mathfrak{B}$ of open subsets of $X$ such that each open subset $G \subseteq X$ is a union of subcollections $U \subseteq \mathfrak{B}$. The concept of a base is a fundamental concept in topology: in many problems concerned with open sets of some space it is sufficient to restrict the considerations to its base. A space can have many bases, the largest one of which is the family of all open sets. The minimum of the cardinalities of all bases is called the ''[[Weight of a topological space|weight]]'' of the topological space $X$. In a space of weight $\tau$ there exists an everywhere-dense set of cardinality $\le \tau$. Spaces with a countable base are also referred to as spaces satisfying the [[second axiom of countability]]. The dual concept of a closed base, formed by the complements of the elements of a base, is used in [[compactification]] theory.

A local base of a space $X$ at a point $x \in X$ (a base of the point $x$) is a family $\mathfrak{B}(x)$ of open sets of $X$ with the following property: For any neighbourhood $O_x$ of $x$ it is possible to find an element $V \in \mathfrak{B}(x)$ such that $x \in V \subseteq O_x$. Spaces with a countable local base at every point are also referred to as spaces satisfying the [[first axiom of countability]]. A family $\mathfrak{B}$ of open sets in $X$ is a base if and only if it is a local base of each one of its points $x \in X$.

Let $\mathfrak{m}, \mathfrak{n}$ be cardinal numbers. A base $\mathfrak{B}$ of the space $X$ is called an $\mathfrak{m}$-point base if each point $x \in X$ belongs to at most $\mathfrak{m}$ elements of the family $\mathfrak{B}$; in particular, if $\mathfrak{m} = 1$, the base is called ''disjoint''; if $\mathfrak{m}$ is finite, it is called ''bounded point finite''; and if $\mathfrak{m} = \aleph_0$, it is called ''point countable''.

A base  $  \mathfrak B $
of the space  $  X $
is called  $  \mathfrak m $-
local if each point  $  x \in X $
has a neighbourhood  $  O _ {x} $
intersecting with at most  $  \mathfrak m $
elements of the family  $  \mathfrak B $; 
in particular, if  $  \mathfrak m = 1 $, 
the base is referred to as discrete; if  $  \mathfrak m $
is finite, it is called bounded locally finite; and if  $  \mathfrak m = \aleph _ {0} $, 
it is called locally countable. A base  $  \mathfrak B $
is called an  $  ( \mathfrak n - \mathfrak m ) $-
point base (or an  $  ( \mathfrak n - \mathfrak m ) $-
local base) if it is a union of a set of cardinality  $  \mathfrak n $
of  $  \mathfrak m $-
point ( $  \mathfrak m $-
local) bases; examples are, for  $  \mathfrak n = \aleph _ {0} $, 
$  \sigma $-
disjoint,  $  \sigma $-
point finite,  $  \sigma $-
discrete and  $  \sigma $-
locally finite bases.

These concepts are used mainly in the criteria of metrizable spaces. Thus, a regular space with a countable base, or satisfying the first axiom of countability and with a point countable base, is metrizable; a regular space with a  $  \sigma $-
discrete or  $  \sigma $-
locally finite base is metrizable (the converse proposition is true in the former case only).

A base  $  \mathfrak B $
of the space  $  X $
is called uniform ( $  k $-
uniform) if for each point  $  x \in X $(
each compact subset  $  F $) 
and for each one of the neighbourhoods  $  O _ {x} $(
$  O _ {F} $) 
only a finite number of elements of the base contain  $  x $(
intersect with  $  F $) 
and at the same time intersect with the complement  $  X \setminus  O _ {x} $(
$  X \setminus  O _ {F} $). 
A space  $  X $
is metrizable if and only if it is paracompact with a uniform base (a Kolmogorov or  $  T _ {0} $-
space with a  $  k $-
uniform base).

A base  $  \mathfrak B $
of the space  $  X $
is called regular if for each point  $  x \in X $
and an arbitrary neighbourhood  $  O _ {x} $
of it there exists a neighbourhood  $  O _ {x}  ^  \prime  $
such that the set of all the elements of the base which intersect both with  $  O _ {x}  ^  \prime  $
and  $  X\setminus  O _ {x} $
is finite. An accessible or  $  T _ {1} $-
space is metrizable if and only if it has a regular base.

A generalization of the concept of a base is the so-called  $  \pi $-
base (lattice base), which is a family  $  \mathfrak B $
of open sets in the space  $  X $
such that each non-empty open set in  $  X $
contains a non-empty set from  $  \mathfrak B $, 
i.e.  $  \mathfrak B $
is dense in  $  X $
according to Hausdorff. All bases are  $  \pi $-
bases, but the converse is not true; thus, the set  $  \mathbf Z  ^ {+} $
in the [[Stone–Čech compactification|Stone–Čech compactification]] of the set of natural numbers in  $  \mathbf Z  ^ {+} $
forms only a  $  \pi $-
base.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. [P.S. Uryson] Urysohn,  , ''Works on topology and other fields of mathematics'' , '''1–2''' , Leningrad  (1951)  (In Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,   B.A. Pasynkov,   "An introduction to the theory of topological spaces and general dimension theory" , Moscow  (1973)  (In Russian)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR>
</table>

====Comments====
Besides the notions of a bounded point-finite base and a bounded local-finite base one also uses that of a point-finite base and a local-finite base. A base (or any family of subsets  $  \mathfrak B $) 
is called point finite if every point  $  x $
belongs to finitely many members of  $  \mathfrak B $, 
i.e. if  $  \mathfrak B _ {x} = \{ {B \in \mathfrak B } : {x \in B } \} $
is finite for every  $  x $. 
Note that the families  $  \mathfrak B _ {x} $
can have arbitrary large finite cardinalities, in contrast to the definition of bounded point finiteness, when the cardinalities of  $  \mathfrak B _ {x} $
are bounded by a fixed finite  $  \mathfrak m $. 
Similar remarks apply to local finiteness.